Velocity moment ($M_{i_1\cdots i_r}=\int v_{i_1}\cdots v_{i_r}f{d}^{3}v$)

Integrating a phase-space distribution function against products of particle velocity components produces its velocity moments. The zeroth gives number density, the first gives number flux, and, after multiplying by particle mass, the second gives momentum flux. Replacing $\mathbf{v}$ by $\mathbf{v}-\mathbf{u}$, where $\mathbf{u}$ is the mean velocity, defines central velocity moments; the second is the pressure tensor divided by particle mass.

For unit particle mass, this symmetric third-order tensor transports the pressure tensor in a kinetic fluid description. Its contraction gives the heat flux $q_i=Q_{ijj}/2$. It vanishes for a local Maxwellian, whose odd central velocity moments vanish, but it need not vanish for a general anisotropic or skewed phase-space distribution function.

Taking velocity moments of a kinetic transport equation couples the equation for a moment of order $r$ to a spatial derivative of a moment of order $r+1$. A finite fluid description therefore requires closure. For example, the pressure tensor equation involves the third central velocity moment.